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Uniqueness of the Machian Solution in the Brans-Dicke Theory

A. Miyazaki
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Abstract

Machian solutions of which the scalar field exhibits the asymptotic behavior $ϕ=O(ρ/ω)$ are generally explored for the homogeneous and isotropic universe in the Brans-Dicke theory. It is shown that the Machian solution is unique for the closed and the open space. Such a solution is restricted to one that satisfies the relation $GM/c^{2}a=const$, which is fixed to $π$ in the theory for the closed model. Another type of solution satisfying $ϕ=O(ρ/ω)$ with the arbitrary coupling constant $% ω$ is obtained for the flat space. This solution has the scalar field $% ϕ\propto ρt^{2}$ and also keeps the relation $GM/c^{2}R=const$ all the time. This Machian relation and the asymptotic behavior $ϕ=O(ρ/ω)$ is equivalent to each other in the Brans-Dicke theory.

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A. Miyazaki. 2001-01-29. Uniqueness of the Machian Solution in the Brans-Dicke Theory. https://arxiv.org/abs/gr-qc/0101112

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